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**Application of optimal homotopy asymptotic method to Burger equations.**
*(English)*
Zbl 1271.35063

Summary: We apply optimal homotopy asymptotic method (OHAM) for finding approximate solutions of the Burger’s-Huxley and Burger’s-Fisher equations. The results obtained by proposed method are compared to those of Adomian decomposition method (ADM). As a result it is concluded that the method is explicit, effective, and simple to use.

### MSC:

35Q35 | PDEs in connection with fluid mechanics |

35A35 | Theoretical approximation in context of PDEs |

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\textit{R. Nawaz} et al., J. Appl. Math. 2013, Article ID 387478, 8 p. (2013; Zbl 1271.35063)

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### References:

[1] | C. A. J. Fletcher, “Burgers’ equation: a model for all reasons,” in Numerical Solutions of Partial Differential Equations (Parkville, 1981), J. Noye, Ed., pp. 139-225, North-Holland, Amsterdam, The Netherlands, 1982. · Zbl 0496.76091 |

[2] | D. L. Young, C. M. Fan, S. P. Hu, and S. N. Atluri, “The Eulerian-Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations,” Engineering Analysis with Boundary Elements, vol. 32, no. 5, pp. 395-412, 2008. · Zbl 1244.76096 |

[3] | J. M. Burgers, “Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion,” Transactions of the Royal Dutch Academy of Sciences in Amsterdam, vol. 17, no. 2, pp. 1-53, 1939. · Zbl 0061.45709 |

[4] | J. Satsuma, “Topics in soliton theory and exactly solvable nonlinear equations,” in Proceedings of the Conference on Nonlinear Evolution Equations, Solitons and the Inverse Scattering Transform Held at the Mathematical Research Institute, Oberwolfach, July-August, 1986, M. Ablowitz, B. Fuchssteiner, and M. Kruskal, Eds., p. 342, World Scientific, Singapore, 1987. |

[5] | H. N. A. Ismail, K. Raslan, and A. A. Abd Rabboh, “Adomian decomposition method for Burger’s-Huxley and Burger’s-Fisher equations,” Applied Mathematics and Computation, vol. 159, no. 1, pp. 291-301, 2004. · Zbl 1062.65110 |

[6] | X. Y. Wang and Y. K. Lu, “Exact solutions of the extended Burgers-Fisher equation,” Chinese Physics Letters, vol. 7, no. 4, pp. 145-147, 1990. |

[7] | M. Javidi, “Modified pseudospectral method for generalized Burger’s-Fisher equation,” International Mathematical Forum, vol. 1, no. 29-32, pp. 1555-1564, 2006. · Zbl 1119.35356 |

[8] | D. Kaya and S. M. El-Sayed, “A numerical simulation and explicit solutions of the generalized Burgers-Fisher equation,” Applied Mathematics and Computation, vol. 152, no. 2, pp. 403-413, 2004. · Zbl 1052.65098 |

[9] | P. Chandrasekaran and E. K. Ramasami, “Painleve analysis of a class of nonlinear diffusion equations,” Journal of Applied Mathematics and Stochastic Analysis, vol. 9, no. 1, pp. 77-86, 1996. · Zbl 0852.35119 |

[10] | H. Chen and H. Zhang, “New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation,” Chaos, Solitons & Fractals, vol. 19, no. 1, pp. 71-76, 2004. · Zbl 1068.35126 |

[11] | E. S. Fahmy, “Travelling wave solutions for some time-delayed equations through factorizations,” Chaos, Solitons & Fractals, vol. 38, no. 4, pp. 1209-1216, 2008. · Zbl 1152.35438 |

[12] | L. Jiang, Y.-C. Guo, and S.-J. Xu, “Some new exact solutions to the Burgers-Fisher equation and generalized Burgers-Fisher equation,” Chinese Physics, vol. 16, no. 9, pp. 2514-2522, 2007. |

[13] | V. Marinca and N. Heri\csanu, “Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer,” International Communications in Heat and Mass Transfer, vol. 35, no. 6, pp. 710-715, 2008. |

[14] | V. Marinca, N. Heri\csanu, C. Bota, and B. Marinca, “An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate,” Applied Mathematics Letters, vol. 22, no. 2, pp. 245-251, 2009. · Zbl 1163.76318 |

[15] | V. Marinca, N. Heri\csanu, and I. Neme\cs, “Optimal homotopy asymptotic method with application to thin film flow,” Central European Journal of Physics, vol. 6, no. 3, pp. 648-653, 2008. |

[16] | R. Nawaz, M. N. Khalid, S. Islam, and S. Yasin, “Solution of tenth order boundary value problems using optimal homotopy asymptotic method (OHAM),” Canadian Journal on Computing in Mathematics, Natural Sciences, Engineering & Medicin, vol. 1, no. 2, pp. 37-54, 2010. |

[17] | S. Iqbal, M. Idrees, A. M. Siddiqui, and A. R. Ansari, “Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 2898-2909, 2010. · Zbl 1193.35190 |

[18] | S. Haq, M. Idrees, and S. Isalam, “Application of optimal homotopy asymptotic Method to eighth order boundary value problems,” Journal of Applied Mathematics and Computing, vol. 2, no. 4, pp. 38-47, 2008. |

[19] | M. Idrees, S. Haq, and S. Islam, “Application of optimal homotopy asymptotic method to fourth order boundary value problems,” World Applied Sciences Journal, vol. 9, no. 2, pp. 131-137, 2010. · Zbl 1198.76095 |

[20] | M. Idrees, S. Islam, S. Haq, and S. Islam, “Application of the optimal homotopy asymptotic Method to squeezing flow,” Computers and Mathematics with Applications, vol. 59, no. 12, pp. 3858-3866, 2010. · Zbl 1205.76189 |

[21] | M. Idrees, S. Haq, and S. Islam, “Application of optimal homotopy asymptotic method to special sixth order boundary value problems,” World Applied Sciences Journal, vol. 9, no. 2, pp. 138-143, 2010. · Zbl 1198.76095 |

[22] | X. Y. Wang, Z. S. Zhu, and Y. K. Lu, “Solitary wave solutions of the generalised Burgers-Huxley equation,” Journal of Physics A, vol. 23, no. 3, pp. 271-274, 1990. · Zbl 0708.35079 |

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